3 Problems in Test 2 - Introduction to Quantum Mechanics 1 | PHY 4604, Exams of Physics

Download 3 Problems in Test 2 - Introduction to Quantum Mechanics 1 | PHY 4604 and more Exams Physics in PDF only on Docsity! Name: Exam 2 - PHY 4604 - Fall 2002 Friday, October, 25, 2002 This exam is closed book and notes. You are not allowed (nor will you need) a calculator. Please use the space provided on the exam to do the problems. You may also use the backs of pages if additional space is needed. 1. Short answer section: (a) What is the definition of the Hermitian conjugate of an operator A? (b) What function corresponds to the state |xo〉? (c) What differential equations do the expectation values of the position and momen- tum operators satisfy for the Hamiltonian H = p 2 2m + V (x)? (d) Express the time dependent Heisenberg operator, A(t), in terms of the time in- dependent Schrodinger operator, A. (e) For a discrete set of states what is the completeness condition in Dirac notation? 1 2. Consider the harmonic oscillator hamiltonian, H = p2 2m + 1 2 mω2x2. Because the eigenstates of H are complete, an initial wave function at t = 0 may be written as |ψ(t = 0)〉 = ∞∑ n=0 cn|n〉, where the cn are complex numbers. (a) What is the condition on the cn so that |ψ(t = 0)〉 is normalized? (b) What is |ψ(t)〉 expressed in terms of cn, |n〉, and ω? 2 3. Consider two electrons described by the Hamiltonian H = p21 2m + p22 2m + V (x1) + V (x2), where V (x) = ∞ for x < 0 and x > a; V (x) = 0 for 0 < x < a. Assume that the electrons are in the same spin state. (a) What is the energy, E, and wave function, ψ(x1, x2), of the second excited state? (b) What is the meaning of |ψ(x1, x2)|2? 5 (c) For the state in part (a) what is the probability of finding either one of the electrons at x = xo? (d) Suppose that the particles are bosons instead of fermions. What is the energy, E, and wave function, ψ(x1, x2), of the second excited state? 6